Forward-Backward Splitting Methods for Accretive Operators in Banach Spaces

and Applied Analysis 3 The main difficulties are due to the fact that the inner product structure of a Hilbert space fails to be true in a Banach space. We shall in this paper use the technique of duality maps to carry out certain initiative investigations on splitting methods for accretive operators in Banach spaces. Namely, we will study splitting iterative methods for solving the inclusion problem 1.1 , where A and B are accretive operators in a Banach space X. We will consider the case where A is single-valued accretive and B is possibly multivalued m-accretive operators in a Banach space X and assume that the inclusion 1.1 has a solution. We introduce the following two iterative methods which we call Mann-type and respectively, Halpern-type forward-backward methods with errors and which generate a sequence {xn} by the recursions xn 1 1 − αn xn αn ( J rn xn − rn Axn an bn ) , 1.6 xn 1 αnu 1 − αn ( J rn xn − rn Axn αn bn ) , 1.7 where J r is the resolvent of the operator B of order r i.e., J B r I rB −1 , and {αn} is a sequence in 0, 1 . We will prove weak convergence of 1.6 and strong convergence of 1.7 to a solution to 1.1 in some class of Banach spaces which will be made clear in Section 3. The paper is organized as follows. In the next section we introduce the class of Banach spaces in which we shall study our splitting methods for solving 1.1 . We also introduce the concept of accretive and m-accretive operators in a Banach space. In Section 3, we discuss the splitting algorithms 1.6 and 1.7 and prove their weak and strong convergence, respectively. In Section 4, we discuss applications of both algorithms 1.6 and 1.7 to variational inequalities, fixed points of pseudocontractions, convexly constrained minimization problems, the split feasibility problem, and linear inverse problems. 2. Preliminaries Throughout the paper,X is a real Banach space with norm ‖ ·‖, distance d, and dual spaceX∗. The symbol 〈x∗, x〉 denotes the pairing between X∗ and X, that is, 〈x∗, x〉 x∗ x , the value of x∗ at x. C will denote a nonempty closed convex subset of X, unless otherwise stated, and Br the closed ball with center zero and radius r. The expressions xn → x and xn ⇀ x denote the strong and weak convergence of the sequence {xn}, respectively, and ωw xn stands for the set of weak limit points of the sequence {xn}. The modulus of convexity of X is the function δX ε : 0, 2 → 0, 1 defined by δX ε inf { 1 − ∥∥x y∥∥ 2 : ‖x‖ ∥∥y∥∥ 1,∥∥x − y∥∥ ≥ ε } . 2.1 Recall thatX is said to be uniformly convex if δX ε > 0 for any ε ∈ 0, 2 . Let p > 1. We say that X is p-uniformly convex if there exists a constant cp > 0 so that δX ε ≥ cpε for any ε ∈ 0, 2 . The modulus of smoothness of X is the function ρX τ : R → R defined by ρX τ sup {∥∥x τy∥∥ ∥∥x − τy∥∥ 2 − 1 : ‖x‖ ∥∥y∥∥ 1 } . 2.2 4 Abstract and Applied Analysis Recall that X is called uniformly smooth if limτ → 0ρX τ /τ 0. Let 1 < q ≤ 2. We say that X is q-uniformly smooth if there is a cq > 0 so that ρX τ ≤ cqτ for τ > 0. It is known that X is p-uniformly convex if and only if X∗ is q-uniformly smooth, where 1/p 1/q 1 . For instance, L spaces are 2-uniformly convex and p-uniformly smooth if 1 < p ≤ 2, whereas p-uniformly convex and 2-uniformly smooth if p ≥ 2. The norm of X is said to be the Fréchet differentiable if, for each x ∈ X, lim λ→ 0 ∥∥x λy∥∥ − ‖x‖ λ 2.3 exists and is attained uniformly for all y such that ‖y‖ 1. It can be proved thatX is uniformly smooth if the limit 2.3 exists and is attained uniformly for all x, y such that ‖x‖ ‖y‖ 1. So it is trivial that a uniformly smooth Banach space has a Fréchet differentiable norm. The subdifferential of a proper convex function f : X → −∞, ∞ is the set-valued operator ∂f : X → 2 defined as ∂f x { x∗ ∈ X∗ : 〈x∗, y − x〉 f x ≤ f(y)}. 2.4 If f is proper, convex, and lower semicontinuous, the subdifferential ∂f x / ∅ for any x ∈ intD f , the interior of the domain of f . The generalized duality mapping Jp : X → 2X is defined by Jp x { j x ∈ X∗ : 〈j x , x〉 ‖x‖, ∥∥j x ∥∥ ‖x‖p−1 } . 2.5 If p 2, the corresponding duality mapping is called the normalized duality mapping and denoted by J. It can be proved that, for any x ∈ X, Jp x ∂ ( 1 p ‖x‖ ) . 2.6 Thus we have the following subdifferential inequality, for any x, y ∈ X: ∥∥x y∥∥p ≤ ‖x‖ p〈y, j(x y)〉, j(x y) ∈ Jp ( x y ) . 2.7 In particular, we have, for x, y ∈ X, ∥∥x y∥∥2 ≤ ‖x‖ 2〈y, j(x y)〉, j(x y) ∈ J(x y). 2.8 Some properties of the duality mappings are collected as follows. Proposition 2.1 see Cioranescu 13 . Let 1 < p < ∞. i The Banach space X is smooth if and only if the duality mapping Jp is single valued. ii The Banach space X is uniformly smooth if and only if the duality mapping Jp is singlevalued and norm-to-norm uniformly continuous on bounded sets of X. Abstract and Applied Analysis 5 Among the estimates satisfied by p-uniformly convex and p-uniformly smooth spaces, the following ones will come in handy. Lemma 2.2 see Xu 14 . Let 1 < p < ∞, q ∈ 1, 2 , r > 0 be given. i If X is uniformly convex, then there exists a continuous, strictly increasing and convex function φ : R → R with φ 0 0 such that ∥∥λx 1 − λ y∥∥p ≤ λ‖x‖ λ∥∥y∥∥p −Wp λ φ (∥∥x − y∥∥), x, y ∈ Br , 0 ≤ λ ≤ 1, 2.9 whereWp λ λ 1 − λ 1 − λ λ. ii If X is q-uniformly smooth, then there exists a constant κq > 0 such that ∥∥x y∥∥q ≤ ‖x‖ qJq x , y 〉 κq ∥∥y∥∥q, x, y ∈ X. 2.10and Applied Analysis 5 Among the estimates satisfied by p-uniformly convex and p-uniformly smooth spaces, the following ones will come in handy. Lemma 2.2 see Xu 14 . Let 1 < p < ∞, q ∈ 1, 2 , r > 0 be given. i If X is uniformly convex, then there exists a continuous, strictly increasing and convex function φ : R → R with φ 0 0 such that ∥∥λx 1 − λ y∥∥p ≤ λ‖x‖ λ∥∥y∥∥p −Wp λ φ (∥∥x − y∥∥), x, y ∈ Br , 0 ≤ λ ≤ 1, 2.9 whereWp λ λ 1 − λ 1 − λ λ. ii If X is q-uniformly smooth, then there exists a constant κq > 0 such that ∥∥x y∥∥q ≤ ‖x‖ qJq x , y 〉 κq ∥∥y∥∥q, x, y ∈ X. 2.10 The best constant κq satisfying 2.10 will be called the q-uniform smoothness coefficient of X. For instance [14], for 2 ≤ p < ∞, L is 2-uniformly smooth with κ2 p − 1, and for 1 < p ≤ 2, L is p-uniformly smooth with κp 1 t p−1 p 1 tp 1−p, where tp is the unique solution to the equation ( p − 2)tp−1 (p − 1)tp−2 − 1 0, 0 < t < 1. 2.11 In a Banach space X with the Fréchet differentiable norm, there exists a function h : 0,∞ → 0,∞ such that limt→ 0h t /t 0 and for all x, u ∈ X 1 2 ‖x‖ 〈u,J x 〉 ≤ 1 2 ‖x u‖ ≤ 1 2 ‖x‖ 〈u,J x 〉 h ‖u‖ . 2.12 Recall that T : C → C is a nonexpansive mapping if ‖Tx − Ty‖ ≤ ‖x − y‖, for all x, y ∈ C. From now on, Fix T denotes the fixed point set of T . The following lemma claims that the demiclosedness principle for nonexpansive mappings holds in uniformly convex Banach spaces. Lemma 2.3 see Browder 15 . Let C be a nonempty closed convex subset of a uniformly convex space X and T a nonexpansive mapping with Fix T / ∅. If {xn} is a sequence in C such that xn ⇀ x and I − T xn → y, then I − T x y. In particular, if y 0, then x ∈ Fix T . A set-valued operator A : X → 2 , with domain D A and range R A , is said to be accretive if, for all t > 0 and every x, y ∈ D A , ∥∥x − y∥∥ ≤ ∥∥x − y t u − υ ∥∥, u ∈ Ax, υ ∈ Ay. 2.13 It follows from Lemma 1.1 of Kato 16 thatA is accretive if and only if, for each x, y ∈ D A , there exists j x − y ∈ J x − y such that 〈 u − υ, j(x − y)〉 ≥ 0, u ∈ Ax, υ ∈ Ay. 2.14 6 Abstract and Applied Analysis An accretive operator A is said to be m-accretive if the range R I λA X for some λ > 0. It can be shown that an accretive operator A is m-accretive if and only if R I λA X for all λ > 0. Given α > 0 and q ∈ 1,∞ , we say that an accretive operator A is α-inverse strongly accretive α-isa of order q if, for each x, y ∈ D A , there exists jq x − y ∈ Jq x − y such that 〈 u − υ, jq ( x − y)〉 ≥ α‖u − υ‖, u ∈ Ax, υ ∈ Ay. 2.15 When q 2, we simply say α-isa, instead of α-isa of order 2; that is, T is α-isa if, for each x, y ∈ D A , there exists j x − y ∈ J x − y such that 〈 u − υ, j(x − y)〉 ≥ α‖u − υ‖, u ∈ Ax, υ ∈ Ay. 2.16 Given a subsetK of C and a mapping T : C → K, recall that T is a retraction of C onto K if Tx x for all x ∈ K. We say that T is sunny if, for each x ∈ C and t ≥ 0, we have T tx 1 − t Tx Tx, 2.17 whenever tx 1 − t Tx ∈ C. The first result regarding the existence of sunny nonexpansive retractions onto the fixed point set of a nonexpansive mapping is due to Bruck. Theorem 2.4 see Bruck 17 . If X is strictly convex and uniformly smooth and if T : C → C is a nonexpansive mapping having a nonempty fixed point set Fix T , then there exists a sunny nonexpansive retraction of C onto Fix T . The following technical lemma regarding convergence of real sequences will be used when we discuss convergence of algorithms 1.6 and 1.7 in the next section. Lemma 2.5 see 18, 19 . Let {an}, {cn} ⊂ R , {αn} ⊂ 0, 1 , and {bn} ⊂ R be sequences such that an 1 ≤ 1 − αn an bn cn, ∀n ≥ 0. 2.18 Assume ∑∞ n 0 cn < ∞. Then the following results hold: (i) If bn ≤ αnM whereM ≥ 0, then {an} is a bounded sequence. (ii) If ∑∞ n 0 αn ∞ and lim supn→∞bn/αn ≤ 0, then liman 0. 3. Splitting Methods for Accretive Operators In this section we assume that X is a real Banach space and C is a nonempty closed subset of X. We also assume that A is a single-valued and α-isa operator for some α > 0 and B is an m-accretive operator in X, with D A ⊃ C and D B ⊃ C. Moreover, we always use Jr to denote the resolvent of B of order r > 0; that is, Jr ≡ J r I rB −1. 3.1 Abstract and Applied Analysis 7 It is known that the m-accretiveness of B implies that Jr is single valued, defined on the entire X, and firmly nonexpansive; that is, ∥Jrx − Jry ∥∥ ≤ ∥∥s(x − y) 1 − s Jrx − Jry )∥∥, x, y ∈ X, 0 ≤ s ≤ 1. 3.2and Applied Analysis 7 It is known that the m-accretiveness of B implies that Jr is single valued, defined on the entire X, and firmly nonexpansive; that is, ∥Jrx − Jry ∥∥ ≤ ∥∥s(x − y) 1 − s Jrx − Jry )∥∥, x, y ∈ X, 0 ≤ s ≤ 1. 3.2 Below we fix the following notation: Tr : Jr I − rA I rB −1 I − rA . 3.


Introduction
Splitting methods have recently received much attention due to the fact that many nonlinear problems arising in applied areas such as image recovery, signal processing, and machine learning are mathematically modeled as a nonlinear operator equation and this operator is decomposed as the sum of two possibly simpler nonlinear operators.Splitting methods for linear equations were introduced by Peaceman and Rachford 1 and Douglas and Rachford 2 .Extensions to nonlinear equations in Hilbert spaces were carried out by Kellogg 3 and Lions and Mercier 4 see also [5][6][7] .The central problem is to iteratively find a zero of Many problems can be formulated as a problem of form 1.1 .For instance, a stationary solution to the initial value problem of the evolution equation ∂u ∂t Fu 0, u 0 u 0 1.2 can be recast as 1.1 when the governing maximal monotone F is of the form F A B 4 .
In optimization, it often needs 8 to solve a minimization problem of the form where f, g are proper lower semicontinuous convex functions from H to the extended real line R : −∞, ∞ , and T is a bounded linear operator on H.As a matter of fact, 1.3 is equivalent to 1.1 assuming that f and g • T have a common point of continuity with A : ∂f and B : T * • ∂g • T .Here T * is the adjoint of T and ∂f is the subdifferential operator of f in the sense of convex analysis.It is known 8, 9 that the minimization problem 1.3 is widely used in image recovery, signal processing, and machine learning.
A splitting method for 1.1 means an iterative method for which each iteration involves only with the individual operators A and B, but not the sum A B. To solve 1.1 , Lions and Mercier 4 introduced the nonlinear Peaceman-Rachford and Douglas-Rachford splitting iterative algorithms which generate a sequence {υ n } by the recursion and respectively, a sequence {υ n } by the recursion Here we use J T λ to denote the resolvent of a monotone operator T ; that is, J T λ I λT −1 .The nonlinear Peaceman-Rachford algorithm 1.4 fails, in general, to converge even in the weak topology in the infinite-dimensional setting .This is due to the fact that the generating operator 2J A λ − I 2J B λ − I for the algorithm 1.4 is merely nonexpansive.However, the mean averages of {u n } can be weakly convergent 5 .The nonlinear Douglas-Rachford algorithm 1.5 always converges in the weak topology to a point u and u J B λ υ is a solution to 1.1 , since the generating operator J A λ 2J B λ − I I − J B λ for this algorithm is firmly nonexpansive, namely, the operator is of the form I T /2, where T is nonexpansive.
There is, however, little work in the existing literature on splitting methods for nonlinear operator equations in the setting of Banach spaces though there was some work on finding a common zero of a finite family of accretive operators 10-12 .
The main difficulties are due to the fact that the inner product structure of a Hilbert space fails to be true in a Banach space.We shall in this paper use the technique of duality maps to carry out certain initiative investigations on splitting methods for accretive operators in Banach spaces.Namely, we will study splitting iterative methods for solving the inclusion problem 1.1 , where A and B are accretive operators in a Banach space X.
We will consider the case where A is single-valued accretive and B is possibly multivalued m-accretive operators in a Banach space X and assume that the inclusion 1.1 has a solution.We introduce the following two iterative methods which we call Mann-type and respectively, Halpern-type forward-backward methods with errors and which generate a sequence {x n } by the recursions where J B r is the resolvent of the operator B of order r i.e., J B r I rB −1 , and {α n } is a sequence in 0, 1 .We will prove weak convergence of 1.6 and strong convergence of 1.7 to a solution to 1.1 in some class of Banach spaces which will be made clear in Section 3.
The paper is organized as follows.In the next section we introduce the class of Banach spaces in which we shall study our splitting methods for solving 1.1 .We also introduce the concept of accretive and m-accretive operators in a Banach space.In Section 3, we discuss the splitting algorithms 1.6 and 1.7 and prove their weak and strong convergence, respectively.In Section 4, we discuss applications of both algorithms 1.6 and 1.7 to variational inequalities, fixed points of pseudocontractions, convexly constrained minimization problems, the split feasibility problem, and linear inverse problems.

Preliminaries
Throughout the paper, X is a real Banach space with norm • , distance d, and dual space X * .The symbol x * , x denotes the pairing between X * and X, that is, x * , x x * x , the value of x * at x. C will denote a nonempty closed convex subset of X, unless otherwise stated, and B r the closed ball with center zero and radius r.The expressions x n → x and x n x denote the strong and weak convergence of the sequence {x n }, respectively, and ω w x n stands for the set of weak limit points of the sequence {x n }.
The modulus of convexity of X is the function δ X ε : 0, 2 → 0, 1 defined by Recall that X is said to be uniformly convex if δ X ε > 0 for any ε ∈ 0, 2 .Let p > 1.We say that X is p-uniformly convex if there exists a constant c p > 0 so that δ X ε ≥ c p ε p for any ε ∈ 0, 2 .The modulus of smoothness of X is the function ρ X τ : R → R defined by Abstract and Applied Analysis Recall that X is called uniformly smooth if lim τ → 0 ρ X τ /τ 0. Let 1 < q ≤ 2. We say that X is q-uniformly smooth if there is a c q > 0 so that ρ X τ ≤ c q τ q for τ > 0. It is known that X is p-uniformly convex if and only if X * is q-uniformly smooth, where 1/p 1/q 1 .For instance, L p spaces are 2-uniformly convex and p-uniformly smooth if 1 < p ≤ 2, whereas p-uniformly convex and 2-uniformly smooth if p ≥ 2.
The norm of X is said to be the Fréchet differentiable if, for each x ∈ X, exists and is attained uniformly for all y such that y 1.It can be proved that X is uniformly smooth if the limit 2.3 exists and is attained uniformly for all x, y such that x y 1.So it is trivial that a uniformly smooth Banach space has a Fréchet differentiable norm.
The subdifferential of a proper convex function f : If f is proper, convex, and lower semicontinuous, the subdifferential ∂f x / ∅ for any x ∈ int D f , the interior of the domain of f.The generalized duality mapping J p : X → 2 X * is defined by

2.5
If p 2, the corresponding duality mapping is called the normalized duality mapping and denoted by J.It can be proved that, for any x ∈ X,

2.6
Thus we have the following subdifferential inequality, for any x, y ∈ X: x y p ≤ x p p y, j x y , j x y ∈ J p x y .

2.7
In particular, we have, for x, y ∈ X, x y 2 ≤ x 2 2 y, j x y , j x y ∈ J x y .

2.8
Some properties of the duality mappings are collected as follows.
i The Banach space X is smooth if and only if the duality mapping J p is single valued.
ii The Banach space X is uniformly smooth if and only if the duality mapping J p is singlevalued and norm-to-norm uniformly continuous on bounded sets of X.
Among the estimates satisfied by p-uniformly convex and p-uniformly smooth spaces, the following ones will come in handy.Lemma 2.2 see Xu 14 .Let 1 < p < ∞, q ∈ 1, 2 , r > 0 be given.
i If X is uniformly convex, then there exists a continuous, strictly increasing and convex function ϕ : R → R with ϕ 0 0 such that where ii If X is q-uniformly smooth, then there exists a constant κ q > 0 such that x y q ≤ x q q J q x , y κ q y q , x,y ∈ X.

2.10
The best constant κ q satisfying 2.10 will be called the q-uniform smoothness coefficient of X.For instance [14], for 2 ≤ p < ∞, L p is 2-uniformly smooth with κ 2 p − 1, and for 1 < p ≤ 2, L p is p-uniformly smooth with κ p 1 t p−1 p 1 t p 1−p , where t p is the unique solution to the equation In a Banach space X with the Fréchet differentiable norm, there exists a function h : 0, ∞ → 0, ∞ such that lim t → 0 h t /t 0 and for all x, u ∈ X

2.12
Recall that T : C → C is a nonexpansive mapping if Tx − Ty ≤ x − y , for all x, y ∈ C. From now on, Fix T denotes the fixed point set of T .The following lemma claims that the demiclosedness principle for nonexpansive mappings holds in uniformly convex Banach spaces.Lemma 2.3 see Browder 15 .Let C be a nonempty closed convex subset of a uniformly convex space X and T a nonexpansive mapping with Fix T / ∅.If {x n } is a sequence in C such that x n x and I − T x n → y, then I − T x y.In particular, if y 0, then x ∈ Fix T .
A set-valued operator A : X → 2 X , with domain D A and range R A , is said to be accretive if, for all t > 0 and every x, y ∈ D A ,

2.13
It follows from Lemma 1.1 of Kato 16 that A is accretive if and only if, for each x, y ∈ D A , there exists j x − y ∈ J x − y such that u − υ, j x − y ≥ 0, u ∈ Ax, υ ∈ Ay.

2.14
An accretive operator A is said to be m-accretive if the range R I λA X for some λ > 0. It can be shown that an accretive operator A is m-accretive if and only if R I λA X for all λ > 0.
Given α > 0 and q ∈ 1, ∞ , we say that an accretive operator A is α-inverse strongly accretive α-isa of order q if, for each x, y ∈ D A , there exists j q x − y ∈ J q x − y such that u − υ, j q x − y ≥ α u − υ q , u ∈ Ax, υ ∈ Ay.

2.15
When q 2, we simply say α-isa, instead of α-isa of order 2; that is, Given a subset K of C and a mapping T : C → K, recall that T is a retraction of C onto K if Tx x for all x ∈ K.We say that T is sunny if, for each x ∈ C and t ≥ 0, we have The first result regarding the existence of sunny nonexpansive retractions onto the fixed point set of a nonexpansive mapping is due to Bruck.The following technical lemma regarding convergence of real sequences will be used when we discuss convergence of algorithms 1.6 and 1.7 in the next section.

Splitting Methods for Accretive Operators
In this section we assume that X is a real Banach space and C is a nonempty closed subset of X.We also assume that A is a single-valued and α-isa operator for some α > 0 and B is an m-accretive operator in X, with D A ⊃ C and D B ⊃ C.Moreover, we always use J r to denote the resolvent of B of order r > 0; that is,

3.1
Abstract and Applied Analysis 7 It is known that the m-accretiveness of B implies that J r is single valued, defined on the entire X, and firmly nonexpansive; that is, Below we fix the following notation: Proof.From the definition of T r , it follows that

3.4
This lemma alludes to the fact that in order to solve the inclusion problem 1.1 , it suffices to find a fixed point of T r .Since T r is already "split," an iterative algorithm for T r corresponds to a splitting algorithm for 1.1 .However, to guarantee convergence weak or strong of an iterative algorithm for T r , we need good metric properties of T r such as nonexpansivity.To this end, we need geometric conditions on the underlying space X see Lemma 3.3 .Lemma 3.2.Given 0 < s ≤ r and x ∈ X, there holds the relation 3.5 Proof.Note that x−T r x /r −Ax ∈ B T r x .By the accretivity of B, we have j s,r ∈ J T s x−T r x such that It turns out that

8 Abstract and Applied Analysis
This along with the triangle inequality yields that

3.8
We notice that though the resolvent of an accretive operator is always firmly nonexpansive in a general Banach space, firm nonexpansiveness is however insufficient to estimate useful bounds which are required to prove convergence of iterative algorithms for solving nonlinear equations governed by accretive operations.To overcome this difficulty, we need to impose additional properties on the underlying Banach space X.Lemma 3.3 below establishes a sharper estimate than nonexpansiveness of the mapping T r , which is useful for us to prove the weak and strong convergence of algorithms 1.6 and 1.7 .Lemma 3.3.Let X be a uniformly convex and q-uniformly smooth Banach space for some q ∈ 1, 2 .Assume that A is a single-valued α-isa of order q in X.Then, given s > 0, there exists a continuous, strictly increasing and convex function φ q : R → R with φ q 0 0 such that, for all x, y ∈ B s , where κ q is the q-uniform smoothness coefficient of X (see Lemma 2.2).
Proof.Put x x − rAx and y y − rAy.Since x − J r x /r ∈ B J r x , it follows from the accretiveness of B that

3.10
Since x, y ∈ B s , by the accretivity of A it is easy to show that there exists t > 0 such that x − y ∈ B t ; hence, J r x − J r y ∈ B t for J r is nonexpansive.Now since X is uniformly convex, we can use Lemma 2.2 to find a continuous, strictly increasing and convex function ϕ : R → R , with ϕ 0 0, satisfying

3.11
Abstract and Applied Analysis 9 where the last inequality follows from the nonexpansivity of the resolvent J r .Letting φ q ϕ/2 q and combining 3.10 and 3.11 yield T r x − T r y q ≤ x − y q − φ q I − J r x − I − J r y .

3.12
On the other hand, since X is also q-uniformly smooth and A is α-isa of order q, we derive that

3.13
Finally the required inequality 3.9 follows from 3.12 and 3.13 .
Remark 3.4.Note that from Lemma 3.3 one deduces that, under the same conditions, if r ≤ αq/κ q 1/ q−1 , then the mapping T r is nonexpansive.

Weak Convergence
Mann's iterative method 20 is a widely used method for finding a fixed point of nonexpansive mappings 21 .We have proved that a splitting method for solving 1.1 can, under certain conditions, be reduced to a method for finding a fixed point of a nonexpansive mapping.It is therefore the purpose of this subsection to introduce and prove its weak convergence of a Mann-type forward-backward method with errors in a uniformly convex and q-uniformly smooth Banach space.See 22 for a similar treatment of the proximal point algorithm 23, 24 for finding zeros of monotone operators in the Hilbert space setting.To this end we need a lemma about the uniqueness of weak cluster points of a sequence, whose proof, included here, follows the idea presented in 21, 25 .
Lemma 3.5.Let C be a closed convex subset of a uniformly convex Banach space X with a Fréchet differentiable norm, and let {T n } be a sequence of nonexpansive self-mappings on C with a nonempty common fixed point set F. If x 0 ∈ C and x n 1 : T n x n e n , where ∞ n 1 e n < ∞, then z 1 − z 2 , J y 1 − y 2 0 for all y 1 , y 2 ∈ F and all z 1 , z 2 weak limit points of {x n }.
Proof.We first claim that the sequence {x n } is bounded.As a matter of fact, for each fixed p ∈ F and any n ∈ N,

3.14
As ∞ n 1 e n < ∞, we can apply Lemma 2.5 to find that lim n → ∞ x n − p exists.In particular, {x n } is bounded.
Let us next prove that, for every y 1 , y 2 ∈ F and 0 < t < 1, the limit where 3.17 By nonexpansivity, we have that

3.20
Let K be a closed bounded convex subset of X containing {x n } and {y 1 , y 2 }.A result of Bruck 26 assures the existence of a strictly increasing continuous function g : 0, ∞ → 0, ∞ with g 0 0 such that for all U : K → X nonexpansive, x, y ∈ K and 0 ≤ t ≤ 1. Applying 3.21 to each S n,m , we obtain

3.22
Now since lim n → ∞ x n − y 1 exists,

3.25
Hence the limit 3.15 exists.If we replace now x and u in 2.12 with y 1 − y 2 and t x n − y 1 , respectively, we arrive at

3.26
Since the lim n → ∞ x n − y 1 exists, we deduce that

3.27
where lim t → 0 o t /t 0. Consequently, we deduce that lim sup

3.28
Setting t tend to 0, we conclude that lim n → ∞ x n − y 1 , J y 1 − y 2 exists.Therefore, for any two weak limit points z 1 and z 2 of {x n }, Theorem 3.6.Let X be a uniformly convex and q-uniformly smooth Banach space.Let A : X → X be an α-isa of order q and B : X → 2 X an m-accretive operator.Assume that S A B −1 0 / ∅.We define a sequence {x n } by the perturbed iterative scheme where J r n I r n B −1 , {a n }, {b n } ⊂ X, {α n } ⊂ 0, 1 , and Then {x n } converges weakly to some x ∈ S. Proof

3.32
Therefore, condition i implies e n < ∞.

3.33
Take z ∈ S to deduce that, as S Fix T n and T n is nonexpansive,

3.34
Due to 3.33 , Lemma 2.5 is applicable and we get that lim n → ∞ x n − z exists; in particular, {x n } is bounded.Let M > 0 be such that x n < M, for all n ∈ N, and let s q M z q−1 .By 2.7 and Lemma 3.3, we have T n x n − x n 0.

3.37
Since lim inf n → ∞ r n > 0, there exists ε > 0 such that r n ≥ ε for all n ≥ 0.Then, by Lemma 3.2, By Lemmas 3.3 and 3.1, T ε is nonexpansive and Fix T ε S / ∅.We can therefore make use of Lemma 2.3 to assure that ω w x n ⊂ S.

3.39
Finally we set U n 1 − α n I α n T n and rewrite scheme 3.31 as x n 1 U n x n e n , 3.40 where the sequence {e n } satisfies ∞ n 1 e n < ∞.Since {U n } is a sequence of nonexpansive mappings with S as its nonempty common fixed point set, and since the space X is uniformly convex with a Fréchet differentiable norm, we can apply Lemma 3.5 together with 3.39 to assert that the sequence {x n } has exactly one weak limit point; it is therefore weakly convergent.

Strong Convergence
Halpern's method 27 is another iterative method for finding a fixed point of nonexpansive mappings.This method has been extensively studied in the literature 28-30 see also the recent survey 31 .In this section we aim to introduce and prove the strong convergence of a Halpern-type forward-backward method with errors in uniformly convex and q-uniformly smooth Banach spaces.This result turns out to be new even in the setting of Hilbert spaces.Theorem 3.7.Let X be a uniformly convex and q-uniformly smooth Banach space.Let A : X → X be an α-isa of order q and B : X → 2 X an m-accretive operator.Assume that S A B −1 0 / ∅.We define a sequence {x n } by the iterative scheme where u ∈ X, J r n I r n B −1 , {a n }, {b n } ⊂ X, {α n } ⊂ 0, 1 , and {r n } ⊂ 0, ∞ .Assume the following conditions are satisfied: Then {x n } converges in norm to z Q u , where Q is the sunny nonexpansive retraction of X onto S.
Proof.Let z Q u , where Q is the sunny nonexpansive retraction of X onto S whose existence is ensured by Theorem 2.4.Let y n be a sequence generated by where we abbreviate T n : J r n I − r n A .Hence to show the desired result, it suffices to prove that y n → z.Indeed, since J r n and I − r n A are both nonexpansive, it follows that

3.43
where L : max 1, αq/κ q 1/ q−1 .According to condition i , we can apply Lemma 2.5 ii to conclude that x n − y n → 0 as n → ∞.
We next show y n → z.Indeed, since S Fix T n and T n is nonexpansive, we have

3.44
Hence, we can apply Lemma 2.5 i to claim that {y n } is bounded.Using the inequality 2.7 with p q, we derive that

3.45
By condition iii , we have some δ > 0 such that for all n ∈ N. Hence, by Lemma 3.3 we get from 3.45 that

3.47
Let us define s n y n − z q for all n ≥ 0. Depending on the asymptotic behavior of the sequence {s n } we distinguish two cases.
Case 1. Suppose that there exists N ∈ N such that the sequence {s n } n≥N is nonincreasing; thus, lim n → ∞ s n exists.Since α n → 0 and e n → 0, it follows immediately from 3.47 that T n y n − y n 0.

3.49
By condition iii , there exists ε > 0 such that r n ≥ ε for all n ≥ 0.Then, by Lemma 3.2, we get T n y n − y n 0.

3.51
Note that from inequality 3.47 we deduce that

3.52
Next we prove that lim sup Equivalently should y n − z 0 , we need to prove that lim sup

3.54
To this end, let z t satisfy z t tu T ε z t .By Reich's theorem 32 , we get z t → Q S u z as t → 0. Using subdifferential inequality, we deduce that

3.56
Then it follows from 3.55 that

3.58
Then, letting t → 0 and noting the fact that the duality map J is norm-to-norm uniformly continuous on bounded sets, we get 3.54 as desired.Due to 3.53 , we can apply Lemma 2.5 ii to 3.52 to conclude that s n → 0; that is, y n → z.
Case 2. Suppose that there exists n 1 ∈ N such that s n 1 ≤ s n 1 1 .Let us define

3.60
Note that the sequence {τ n } is nonincreasing and lim n → ∞ τ n ∞.Moreover, τ n ≤ n and for any n ≥ n 1 see Lemma 3.1 of Maingé 33 for more details .From inequality 3.47 we get

3.63
It turns out that T τ n y τ n − y τ n 0.

3.65
Now repeating the argument of the proof of 3.53 in Case 1, we can get lim sup By the asymptotic regularity of {y τ n } and 3.65 , we deduce that lim n → ∞ y τ n 1 − y τ n 0.

3.67
This implies that lim sup

3.68
On the other hand, it follows from 3.64 that

3.69
Taking the lim sup n → ∞ in 3.69 and using condition i we deduce that lim sup n → ∞ s τ n ≤ 0; hence lim n → ∞ s τ n 0. That is, y τ n − z → 0. Using the triangle inequality, we also get that lim n → ∞ s τ n 1 0 which together with 3.42 guarantees that y n − z → 0.

Applications
The two forward-backward methods previously studied, 3.29 and 3.41 , find applications in other related problems such as variational inequalities, the convex feasibility problem, fixed point problems, and optimization problems.
Throughout this section, let C be a nonempty closed and convex subset of a Hilbert space H.Note that in this case the concept of monotonicity coincides with the concept of accretivity.
Regarding the problem we concern, of finding a zero of the sum of two accretive operators in a Hilbert space H, as a direct consequence of Theorem 3.7, we first obtain the following result due to Combettes 34 .Corollary 4.1.Let A : H → H be monotone and B : H → H maximal monotone.Assume that κA is firmly nonexpansive for some κ > 0 and that Then the sequence {x n } generated by the algorithm

Variational Inequality Problems
A monotone variational inequality problem VIP is formulated as the problem of finding a point x ∈ C with the property: where A : C → H is a nonlinear monotone operator.We shall denote by S the solution set of 4.3 and assume S / ∅.One method for solving VIP 4.3 is the projection algorithm which generates, starting with an arbitrary initial point x 0 ∈ H, a sequence {x n } satisfying where r is properly chosen as a stepsize.If in addition A is κ-inverse strongly monotone ism , then the iteration 4.4 with 0 < r < 2κ converges weakly to a point in S whenever such a point exists.
By 35, Theorem 3 , VIP 4.3 is equivalent to finding a point x so that 0 ∈ A B x, 4.5 where B is the normal cone operator of C. In other words, VIPs are a special case of the problem of finding zeros of the sum of two monotone operators.Note that the resolvent of the normal cone is nothing but the projection operator and that if A is κ-ism, then the set Ω is closed and convex 36 .As an application of the previous sections, we get the following results.
Corollary 4.2.Let A : C → H be κ-ism for some κ > 0, and let the following conditions be satisfied: Then the sequence {x n } generated by the relaxed projection algorithm converges weakly to a point in S.
Corollary 4.3.Let A : C → H be κ-ism and let the following conditions be satisfied: Then, for any given u ∈ C, the sequence {x n } generated by

Fixed Points of Strict Pseudocontractions
An operator T : C → C is said to be a strict κ-pseudocontraction if there exists a constant κ ∈ 0, 1 such that To solve the problem of approximating fixed points for such operators, an iterative scheme is provided in the following result.
Then, for any given u ∈ C, the sequence {x n } generated by the algorithm converges strongly to the point P Fix T u.
Proof.Set A I − T .Hence A is 1 − κ /2-ism.Moreover we rewrite the above iteration as Then, by setting B the operator constantly zero, Corollary 4.3 yields the result as desired.

Convexly Constrained Minimization Problem
Then the sequence {x n } generated by the algorithm Then for any given u ∈ C, the sequence {x n } generated by the algorithm converges strongly to P Ω u whenever such point exists.

Split Feasibility Problem
The split feasibility problem SFP 42 consists of finding a point x satisfying the property: x ∈ C, A x ∈ Q, 4.17 where C and Q are, respectively, closed convex subsets of Hilbert spaces H and K and A : H → K is a bounded linear operator.The SFP 4.17 has attracted much attention due to its applications in signal processing 42 .Various algorithms have, therefore, been derived to solve the SFP 4.17 see 39, 43, 44 and reference therein .In particular, Byrne 43 introduced the so-called CQ algorithm: x

Convexly Constrained Linear Inverse Problem
The constrained linear system Ax b x ∈ C,

4.24
where A : H → K is a bounded linear operator and b ∈ K, is called convexly constrained linear inverse problem cf.47 .A classical way to deal with this problem is the well-known projected Landweber method see 40 : x n 1 P C x n − λA * Ax n − b , 4.25 where 0 < λ < 2ν with ν 1/ A 2 .A counterexample in 8, Remark 5.12 shows that the projected Landweber iteration converges weakly in infinite-dimensional spaces, in general.
To get strong convergence, Eicke introduced the so-called damped projection method see 47 .In what follows, we present another algorithm with strong convergence, for solving 4.24 .
Then, for any given u ∈ H, the sequence {x n } generated by the algorithm converges strongly to a solution to problem 4.24 whenever it exists.
Proof.This is an immediate consequence of Corollary 4.8 by taking Q {b}.

Theorem 2 . 4
seeBruck 17  .If X is strictly convex and uniformly smooth and if T : C → C is a nonexpansive mapping having a nonempty fixed point set Fix T , then there exists a sunny nonexpansive retraction of C onto Fix T .

Corollary 4 . 5 .
Let T : C → C be strictly κ-pseudocontractive with a nonempty fixed point set Fix T .Suppose that y 1 − y 2 3.15 exists.To see this, we set S n,m T n m−1 T n m−2 • • • T n which is nonexpansive.It is to see that we can rewrite {x n } in the manner Proof.It suffices to show that κA is firmly nonexpansive if and only if A is κ-inverse strongly monotone.This however follows from the following straightforward observation: Assume that f : H → R is convex and differentiable with 1/κ -Lipschitz continuous gradient ∇f.Assume also that n 1 P C x n − λA * I − P Q Ax n , 4.18where 0 < λ < 2ν with ν 1/ A 2 .To solve the SFP 4.17 , it is very useful to investigate the following convexly constrained minimization problem CCMP : Generally speaking, the SFP 4.17 and CCMP 4.19 are not fully equivalent: every solution to the SFP 4.17 is evidently a minimizer of the CCMP 4.19 ; however a solution to the CCMP 4.19 does not necessarily satisfy the SFP 4.17 .Further, if the solution set of the SFP 4.17 is nonempty, then it follows from 45, Lemma 4.2 thatC ∩ ∇f −1 0 / ∅, 4.21where f is defined by 4.20 .As shown byXu 46, the CQ algorithm need not converge strongly in infinite-dimensional spaces.We now consider an iteration process with strong convergence for solving the SFP 4.17 .Assume that the SFP 4.17 is consistent, and let S be its nonempty solution set.Assume also thati lim n → ∞ α n 0 and ∞ n 1 α n ∞; ii 0 < lim n → ∞ λ n ≤ lim sup n → ∞ λ n < 2ν.Then for any given u ∈ C, the sequence x n generated by the algorithmx n 1 α n u 1 − α n P C x n − λ n A * I − P Q Ax n 4.22converges strongly to the solution P S u of the SFP 4.17 .Proof.Let f be defined by 4.19 .According to 39, page 113 , we have ∇f A * I − P Q A, 4.23 which is 1/ν -Lipschitz continuous with ν 1/ A 2 .Thus Corollary 4.7 applies, and the result follows immediately.Remark 4.9.Corollary 4.8 improves and recovers the result of 44, Corollary 3.7 , which uses the additional condition ∞ n 1 |α n 1 − α n | < ∞, condition i , and the special case of condition ii where λ n ≡ λ for all n ∈ N.